## Theories 1-4

Guide written by **LE★Baldy & Snaeky**. Contributions from **the Amazing Community**.

This guide is currently undergoing change. Keep in mind, **strategies may change**.

Feel free to use the glossary as needed.

### Theory basics #

Publications are equivalent to prestiges for \(f(t)\) so don’t be afraid touse them. However, the best publication multipliers vary from theory to theory and willchange over time. If you are close to a multiplier you want, turn off autobuyerand let \(\rho\) increase without buying upgrades for a faster short-term increasebefore the publication (turn on after you publish). This is known and referenced as “cruising”.Total \(τ\), found in the equation or at the top of the screen, is a multiplicativecombination of all \(τ\) from each theory.

**Don’t be afraid to skip getting all milestones to work on the next or abetter theory.**

###### Note: If you see # → [# → # → #] → # in the milestone route of a theory, this is the section that has an active strategy tied to it.

### Graduation routing #

Remember to follow our routing advice from Introduction to Graduation.

5k | → | 5.2k | → | 5.6k | → | 5.8k | → | 6k |

6k | → | 7k | → | 8k | ||||

8k | → | 8.4k | → | 8.6k | → | 8.8k | → | 9k |

### Theory 1 (20σ / 5k) #

#### T1 Overview #

In mathematics, a recurrence relation is an equation that relies on aninitial term and a previous term to change.We start with the current tick’s term, \(ρ_{n}\), and a constant add-on toobtain the value of the next tick, \(ρ_{n+1}\). This gives us an equationequivalent to \(ρ=at+constant\), with a changing value \(a\) and a constantthat is the initial value of 1. Later when we add the \(c_{3}ρ_{n-1}^{0.2}\) term,this is now saying that we are now adding each tick the value of \(ρ\) fromthe previous tick ago with a constant \(c_{3}\) put to the power of \(0.2\). Thisis the same with the next term \(c_{4}ρ_{n-2}^{0.3}\), with the value of \(ρ\) two ticksago and a multiplier \(c_4\) put to the power \(0.3\). When we multiply the\(c_1c_2\) term by the term \(1+ln(ρ)/100\) changing the constant addition tobeing based on the value of \(ρ\) from the previous tick with the value of\(1+ln(ρ)/100\). The final milestone upgrade raises the exponent of \(c_1\) from\(1.00\) to \(1.05\) to \(1.10\) to \(1.15\).

This theory also has its adjusted tickspeed calculated by \(q_{1}*q_{2}\). Thislengthens the normal tick length of \(0.1/sec\) to that value which speedsup the theory.

#### T1 formula #

##### Initial

\[ρ_{n+1} = ρ_n + c_1c_2\]

##### First milestone

\[ρ_{n+1} = ρ_n + c_1c_2 + c_3ρ_{n-1}^{0.2}\]

##### Second milestone

\[ρ_{n+1} = ρ_n + c_1c_2 + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}\]

##### Third milestone

\[ρ_{n+1} = ρ_n + c_1c_2 \left( 1+\frac{ln(ρ_n)}{100} \right) \\ + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}\]

##### Fourth to Sixth milestone

\[ρ_{n+1} = ρ_n + c_1^{1.15}c_2 \left( 1+\frac{ln(ρ_n)}{100} \right) \\ + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}\]

#### T1 strategy #

The publication multiplier has no optimal fit, as it fluctuates a lot,but here is known: 4-6 to start; 3-4 between 1e100 and 1e150; thepublication multiplier oscillates between 2.5 and 5 past e150. Once youget your first milestone, you can turn off \(c_1\) and \(c_2\) until e150 active strat.

The active strat follows but only works when you have all milestonespast e150. T1 is the only theory where the recent value of \(ρ\)influences the rate of change of \(ρ\) therefore buying a variable assoon as you can afford it will slow your progress. Lategame, buyingupgrades immediately will slow you more than the benefit of the upgradebecause \(c_3\) and \(c_4\) dominate. If the next level costs \(10ρ\)and you have \(11ρ\), buying that level will reduce \(ρ_{n+1}\) to \(1\). This reduces your \(ρ_{n+1}\) by roughly a factor of \(10\).There are \(3\) terms that influence the rate of change of \(ρ\), and all are affected by the previous state of \(ρ\). The active strategy around this is known as T1Ratio. The values in the chart found here are to beonly used when you are past \(e150 τ\) and max milestones. They represent how to purchase each variable based on the state of the theory at the time of purchase.

Note: If you are not doing the active strat, then simply turn off \(c_1\) and \(c_2\) after milestone 1 (e25τ) and autobuy rest until ee6k.

**The video below is only good for early tau between 1e150 and 1e250.**

#### T1 milestone route #

0/0/1 | → | 0/0/1/1 | → | 0/1/1/1 |

0/1/1/1 | → | 3/1/1/1 |

Or | ||||||

3 | → | 4 | → | 2 | ||

2 | → | 1 x3 |

### Theory 2 (25σ / 6k) #

#### T2 Overview #

This second theory is focusing on derivatives. Derivatives inmathematics are the rate of change of the function they are thederivative of. For the case of \(q_1\) and \(q_2\), \(q_2\) isthe derivative of \(q_1\). This follows the power rule for derivatives:

In simpler terms, it works similar to how\(x_i\) upgrades work for \(f(t)\) equation with continuous additionof the previous \(term*dt\) to the next \(x_{i+1}\) term, but withcontinuous addition of \(q_i*dt\) to the term above \(q_{i-1}\).These two values of \(r_1\) and \(q_1\) are multiplied to produce the derivativeof \(ρ(t)\), shown by Newton’s derivative notation \(\dot{ρ}\). This would give theequation of \(ρ\) to be \(ρ(t+dt)=ρ+\dot{ρ}*dt\). The other milestones besides more \(q\)and \(r\) derivatives increase the exponent of \(q\) and \(r\) respectively. Thereason why \(q\) and \(r\) derivatives are more powerful long-term than theexponents is that they take time to build up and eventually overtake andkeep increasing \(q_1\) and \(r_1\) while the exponents have a never-changingboost.

#### T2 formula #

##### Initial

\[\dot{q_n}=q_{n+1}*dt\] for n=1

\[\dot{r_k}=r_{k+1}*dt\] for k=1

\[\dot{ρ}=q_1r_1\]

##### First and Second milestones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1

\[\dot{ρ}=q_1r_1\]

##### Third and Fourth milestones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1, 2, 3

\[\dot{ρ}=q_1r_1\]

##### Fifth to Seventh milestones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1, 2, 3

\[\dot{ρ}=q_1^{1.15}r_1\]

##### Eight to Tenth milestones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1, 2, 3

\[\dot{ρ}=q_1^{1.15}r_1^{1.15}\]

#### T2 strategy #

The optimal multiplier is pretty high and is not known before \(e30\). The theory sim will recommend publication multipliers below these values, but the sim’s T2MS does not currently have coasting.The multipliers for active play (which do use coasting) we know at the moment are:

- \(e25\)-\(e100\) is \(1k\) to \(10k\)
- \(e100\)-\(e175\) is \(10k\)-\(100k\)

**For both strategies the milestones are listed in the order X>Y, where X and Y are the milestones as numerically ordered top to bottom in-game, are to be maxed in order from left to right.**

##### Idle

For the idle strategy, you want to prioritize buying milestone levels of 1>2. If you have more than 4 milestones, you will prioritizemilestone 1>2>3>4. You will want to publish atabout 10-100 multiplier before \(e75\) and about a \(1000\) multiplier after \(e75\), but larger multipliers are fine.If possible, swap to milestones 3>4>1>2 at end before publishing for an additional boost.

##### Active

The goal of the active strategy is to grow \(q_1\) and \(r_1\) asmuch as possible while being able to take advantage of the exponentmilestones too, yeilding a large boost from that growth. The active for T2 is on a 50-second cycle between two milestone sets: 10 seconds forexponent priority (Milestones 3 and 4) and 40 seconds for derivative priority (Milestones 1 and 2) . You will start a publication with exponent priority as the cost of the variables gained from milestones 1 and 2 aretoo large for you to get right away. When you can afford them, you willstart the cycle. The full cycle is listed below:

**1-3 Milestones**

3>4 (10s) → 1 (40s) → 3>4 (10s) → 2 (40s) →

repeat → coast and publish

**4+ Milestones**

3>4>1>2 (10s) → 1>2>3>4 (40s) →

3>4>1>2 (10s) → 2>1>3>4 (40s) →

repeat → coast and publish

Past \(e175\), the active strat will become exponentially lesseffective. At \(e250\), you would start to idle T2 overnight only.Until you have over \(1e350\tau\) from theory 2, this is the best theoryto run idle overnight.

When you get to Theory 3 at ee7k, move on to pushing Theory 3 when active and running T2 overnight. The above is simply an option if you rather not work on T3 now.

#### T2 milestone route #

2/0/0/0 | → | 2/2/0/0 | → | 2/2/3/0 |

2/2/3/0 | → | 2/2/3/3 |

Or | ||||||

1 x2 | → | 2 x2 | → | 3 x3 | ||

3 x3 | → | 4 x3 |

### Theory 3 (30σ / 7k) #

#### T3 Overview #

The basis of this theory and understanding how it works is based onmatrix multiplication. Below I have put a color-coded image to displayhow matrix multiplication works.

This gives the basis for why certain upgrades are more powerful thanothers. The exponents on \(b_1\), \(b_2\), and \(b_3\)all directly affect \(ρ_1\) production which is used for \(\tau\). An extradimension roughly gives \(50%\) more \(\tau\) production as it adds an extra termto the \(ρ_1\) production.

#### T3 strategy #

The optimal publication multiplier is about 2-3 without cruising and 3-4with cruising. If you decide to play actively, there is a form ofexponent swapping strat to be aware of. This is a difficultstrategy because it requires you to notice when a certain thresholdhappens. It happens when the following occurs:

\[c_{11}*b_{1}^{1.05\text{ or }1.1}<c_{12}*b_{2}^{1.05\text{ or }1.1}\]

When this happens swap your exponents from \(b_1\) to \(b_2\) and you will get alittle upgrade boost. It also allows for a slight push of \(ρ_2\) forupgrades to \(b_2\) and \(c_{12}\), but this is a lot less impactful and lessnoticeable. This strategy also works with \(b_3\) and \(c_{13}\) but is usuallynot as common.

If you decide to buy manually, the focus areas are buying \(b_1\), \(b_2\), and \(b_3\) when their cost ise1 lower than \(c_{11}\), \(c_{12}\), and \(c_{13}\) respectively. These all directly boost the productionof \(ρ_1\) which is used for \(\tau\). After this, if you are doing the active exponentswapping strategy described in the previous paragraph, your next focus will be on \(c_{21}\),\(c_{22}\), and \(c_{23}\) as these boost \(b_2\) production which increases the likelihoodfor the exponent swap to occur. This leaves the \(c_{31}\), \(c_{32}\), and \(c_{33}\)upgrades at the lowest priority. If you are not using the exponentswapping strategy from the previous paragraph, then all the remainingupgrades should be bought at equivalent priority.

At the end of any publication, around a 2-3 multiplier, you should turnoff \(b_1\) and \(c_{31}\) as they cost \(ρ_1\). You will cruise until you get to a3-4 multiplier. Publish and turn back on \(ρ_1\) costing variables andrepeat.

###### Commentary

#### T3 milestone route #

Active | ||||
---|---|---|---|---|

0/2/0 | → | 0/2/2 | → | 1/2/2 |

1/2/2/0 | → | 1/2/2/2 |

Or | ||||||

2 x2 | → | 3 x2 | → | 1 | ||

1 | → | 4 x2 |

Idle | ||||
---|---|---|---|---|

0/2/0 | → | 0/2/2 | → | 1/2/2 |

1/2/2/0 | → | 1/2/2/2 |

Or | ||||||

2 x2 | → | 3 x2 | → | 1 | ||

1 | → | 4 x2 |

### Theory 4 (35σ / 8k) #

#### Theory 4 Overview #

Theory 4 is based on Polynomials, which contain terms of the form \(x^a+x^b+x^c\) etc. In this case, instead of ‘x’ it’s ‘q’. The strategies for this theory are quite simple compared to the previous theory, especially late game strategies.

#### Theory 4 Equation Description #

\(\dot{\rho} = c_1^{1.15}c_2 + c_3q + c_4q^2 + c_5q^3 + c_6q^4\)

\(\dot{q} = 8q_1q_2 / (1 + q)\)

The first line statest that the rate of change of rho is the sum of a bunch of polynomial terms. We have a bunch of ‘c’ variables multiplied by ‘q’. We can increase \(q\) by buying \(q_1\) and \(q_2\) upgrades. Note that this is with all milestones. You’ll not have all of these at the beginning.

The second line is more unique. It says that

is proportional to the inverse of \(q\) itself! This means that the more \(q\) we have, the slower \(q\) grows, as

decreases. This means that \(q_1\) and \(q_2\) are not as strong as they first appear. However, we still want to buy them in general unless stated otherwise as slow growth is better than no growth.

For the more mathematically observant reader, we may integrate the \(\dot{q}\) equation and conclude that \(q\) is proportional to the square root of time. This means that even though \(\dot{q}\) grows slower with increasing \(q\), there is theoretically no finite limit on the maximum value of \(q\).

#### Theory Variable Description #

Approximate variable strengths on \(\dot\rho\) with all milestones are as follows:

Brief Description | ||
---|---|---|

c_{1} | About 7% increase on the | |

c_{2} | Doubles the | |

c_{3} | Doubles the | |

c_{4} | Doubles the | |

c_{5} | Doubles the | |

c_{6} | Doubles the | |

q_{1} | About 7% increase on | |

q_{2} | Doubles the instantaneous value of |

#### Theory 4 strategy #

The strengths of each variable are as follows:

__Early game (before 14k ft)__

\(c_6\) > \(c_5\) > \(c_4\) > \(q_2\) > \(c_2\) > \(q_1\) > \(c_3\) > \(c_1\)

__From 14k ft to mid-late game (about e350+ T4)__

\(c_2\) > \(c_3\) > \(q_2\) > \(c_1\) > \(q_1\) > everything else

__From e350+ T4 to end game__

\(c_3\) > \(q_2\) > \(q_1\) > everything else

##### Idle

T4 is quite idle friendly compared to T3 and T1. Here are some simple idle strategies for T4:

__Start to e25__

Autobuy

__e25 to e175__

Get the ‘Add the term’ milestones. Prioritise these ones first until maximum. Now we autobuy

When we unlock

__e175 to endgame__

Simply autobuy

##### Semi-Idle

There’s no strategic difference between semi-idle and idle for this theory. The main difference is with semi-idle, we would publish more often since we check the game more often. We wouldn’t overshoot the optimal multiplier as much.

##### Active

T4 active is more involved. However it is not as demanding as T3 or T1 active.

__Start to e75__

Autobuy

__e75 to e175 OR 14k ft__

Now here is where we can apply some more advanced strategies. Consider that the

- Do the same strategy as before until we reach our previous publication point.
- Take point(s) out of the
exponent milestones and unlock all the terms (the first milestone). We should now have access to${c}_{1}$ .${c}_{6}$ - Autobuy
,${c}_{4}$ ,${c}_{5}$ ,${c}_{6}$ .${q}_{2}$ - If you want to optimise a bit more, you can buy
until its cost exceed about 15% of${q}_{1}$ . Otherwise it’s ok to also autobuy${q}_{2}$ .${q}_{1}$ - DO NOT autobuy
,${c}_{1}$ ,${c}_{2}$ .${c}_{3}$ - Publish at about 10-20. Once published, remember to take out the milestone point and put it back into the
exponent to repeat step 1.${c}_{1}$

If done right, this strategy is significantly faster than the idle strategies above. The logic with this strategy is the

__e175 OR 14k ft to ~e300 T4__

We will do the exact same strategy as in the #start to e75 section above. This is because

__~e300 to endgame__

At this point the

#### T4 milestone route #

3/0/0 | → | 3/0/3 | → | 3/1/3 |

Or | ||||||

1 x3 | → | 3 x3 | → | 2 |

### Theory tier list (Pre-9k+) #

Before you reach 9k, these are the recommended values for each theory.You may not hit the values and have a different distribution, but work on getting these theories up to these values later. This list is in order of priority.

Approximate Tau | |
---|---|

T2 | e240-e300 τ |

T1 | e205-e215 τ |

T3 | e150 τ |

T4 | e150 τ |